Problems to Ponder (December edition)

Welcome to the November edition of Problems to Ponder! This month’s problems have been curated by Michael Pruner, president of the British Columbia Association of Mathematics Teachers (BCAMT). The tasks are released on a weekly basis through the BCAMT listserv, and are also shared via Twitter (@BCAMT) and on the BCAMT website. This post features only a subset of the problems shared by Michael last month – head to the BCAMT website for the full set!

I am calling these problems ‘competency tasks’ because they seem to fit quite nicely with the curricular competencies in the British Columbia revised curriculum. They are non-content based, so that all students should be able to get started and investigate by drawing pictures, making guesses, or asking questions. When possible, extensions are provided so that you can keep your students in flow during the activity. Although they may not fit under a specific topic for your course, the richness of the mathematics comes out when students explain their thinking or show creativity in their solution strategies.

I think it would be fun and more valuable for everyone if we shared our experiences with the tasks. Take pictures of students’ work and share how the tasks worked with your class through the BCAMT listserv [which currently connects nearly one thousand educators from across the province, country, and even the world! –Ed.] so that others may learn from your experiences.

Intermediate and Secondary Tasks (Grades 4-12)

November 6, 2016

Dragon FractalImagine a long strip of paper folded in the same direction once, twice, and then a third time. When the strip is unfolded, how many creases will be on the paper? In what directions will the creases be pointing? What about n folds?

Extensions: When the paper is unfolded, and the creases are made to equal 90°, what do you notice in the shapes?

November 13, 2016

CottagesA circular road is 27 km long. On this road are six cottages, owned by 6 friends. The friends visit each other a lot, and they have noticed that every whole number from 1 to 26 (inclusive) is accounted for at least once when they calculate the distances from one cottage to another. Of course, the friends can walk in either direction as required. Your task is to place these cottages at distances that will fulfill these conditions.

A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?

If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

What creatures could there have been at the park? Which combinations of creatures show the number of legs counted? Show more than one combination.

November 27, 2016

Watermelon SeedsOne hot day, my dad cut a slice of watermelon for me to eat. I counted 13 (change number to meet the needs of the students—e.g., 23 or 33) black and white seeds in the slice. There were more black than white seeds. How many of each kind of seed might there have been?

Michael Pruner is the current president of the British Columbia Association of Mathematics Teachers (BCAMT) and a full-time mathematics teacher at Windsor Secondary School in North Vancouver. He teaches using the Thinking Classroom model where students work collaboratively on tasks to develop both their mathematical competencies and their understanding of the course content.